**1. (Problem 4.37 from Proakis 5th Edition)**

This problem deals with the characteristics of a DPSK signal.

- Suppose we wish to transmit the data sequence 1 1 0 1 0 0 0 1 0 1 1 0 by binary DPSK. Let $s(t)= A cos(2 \pi f_c t + \theta)$ represent the transmitted signal in any signaling interval of duration $T$. Give the phase of the transmitted signal for the data sequence. Begin with $\theta= 0$ for the phase of the first bit to be transmitted.

- If the data sequence is uncorrelated, determine and sketch the power density spectrum of the signal transmitted by DPSK.

**2. (Problem 3.19 from Proakis 5th Edition)**

The information sequence $\{a_n\}^{\infty}_{n=-\infty}$ is a sequence of iid random variables, each taking values $+1$ and $-1$ with equal probability. This sequence is to be transmitted at baseband by a biphase coding scheme, described by

where g(t) is shown in Figure P3.19.

- Find the power spectral density of $s(t)$.

- Assume that it is desirable to have a zero in the power spectrum at $f=1/T$. To this end, we use a precoding scheme by introducing $b_n=a_n + k a_{n-1}$, where $k$ is some constant, and then transmit the $\{b_n\}$ sequence using the same $g(t)$. Is it possible to choose $k$ to produce a frequency null at $f=1/T$? If yes, what are the appropriate values and the resulting power spectrum?

- Now assume we want to have zeros at all multiples of $f_0=1/4T$. Is it possible to have these zeros with an appropriate choice of $k$ in the previous part? If not, then what kind of precoding do you suggest to achieve the desired result?

**3. (Problem 3.22 from Proakis 5th Edition)**

A digital signaling scheme is defined as

where $u(t)= \Lambda(t/2T)$,

(3)and each $(a_n, b_n)$ pair is independent from the others and is equally likely to take any of the three values $(0, 1)$, $(\sqrt{3}/2, -1/2)$, and $(-\sqrt{3}/2, -1/2)$.

- Determine the lowpass equivalent of the modulated signal. Determine the in-phase and quadrature components.

- Determine the power spectral density of the lowpass equivalent signal; form this determine the power spectral density of the modulated signal.

- By employing a precoding scheme of the form

where $\alpha$ is in general a complex number, and transmitting the signal

(5)we want to have a lowpass signal that has no dc component. Is it possible to achieve this goal by an appropriate choice of $\alpha$? If yes, find this value.